In this chapter we will understand the concept of cube and learn the formula to calculate the volume and surface area of cube.
At the end of the chapter, some problems are also provided for practice.
What is a cube ?
Cube is a three – dimensional figure in which all sides are of equal length and vertices of all sides measure 90 degree.
Some properties of cube are;
(a) All sides are equal
(b) All the side angle measure 90 degree
(c) It has 6 faces, 12 edges and 8 vertices
(d) All the face of the cube is in the form of square
(e) Opposite face of cubes are parallel to each other
Volume of cube
The total space enclosed by the cube is called volume of cube.
If ” a ” is the length of one side of the cube, then formula for volume of cube is given as;
Volume = a^{3}
We get volume in cubic units ( \mathtt{cm^{3} \ or\ m^{3} \ etc..} )
Surface area of cube
The total area taken to form the given cube is called surface area of cube.
We know that cube is made of joining 6 squares. To calculate the surface area of cube, find the area of one cube and then multiply by 6.
Let “a” be the length of one side of cube, then formula for surface area is given as;
Surface Area = \mathtt{6a^{2}}
We get surface area in square units ( \mathtt{cm^{2} \ or\ m^{2} \ etc..}
Lateral surface area of cube
Lateral surface area is basically the area of all sides of cube excluding the base and top of the cube.
To calculate the lateral surface area, we will find the area of one square and then multiply it by 4.
Lateral surface area = \mathtt{4a^{2}}
We get lateral surface area in square units.
Perimeter or cube
Perimeter of cube is the length of all its sides.
We know that there are 12 edge in the cube, so the formula for perimeter is given as;
Perimeter = 12 x a
I hope you understood all the above formulas, let us solve some problems related to the concept.
Questions on volume & surface area of cube
Question 01
Find the volume, lateral and total surface area of cube with side length 5 cm.
Solution
\mathtt{Volume\ of\ cube\ =\ a^{3}}\\\ \\ \mathtt{Volume\ of\ cube\ =\ 5^{3} =\ 125\ cm^{3}}\\ \\
Hence, the volume of given cube is 125 cu cm.
\mathtt{Lateral\ surface\ area\ ( cube) \ =\ 4a^{2}}\\\ \\ \mathtt{Lateral\ surface\ area\ ( cube) \ =\ 4( 5)^{2} =100\ cm^{2}}\\ \\
Hence, lateral surface area of given cube is 100 sq. cm
\mathtt{Total\ surface\ area\ ( cube) \ =\ 6a^{2}}\\\ \\ \mathtt{Total\ surface\ area\ ( cube) \ =\ 6( 5)^{2} =150\ cm^{2}} \\ \\
Hence, total surface area of given cube is 150 sq cm.
Question 02
Find the ratio of total surface area to lateral surface area of cube.
Solution
Let us assume the side of cube be “a” units.
\mathtt{Total\ surface\ area\ ( cube) \ =\ 6a^{2}}\\\ \\ \mathtt{Total\ surface\ area\ ( cube) \ =\ 6( 5)^{2} =150\ cm^{2}} \\ \\
Hence, 3 : 2 is the required ratio.
Question 03
Three cubes of 20 cm are placed next to each other. Find the volume of new figure formed.
Solution
Let’s first find the volume of given cube.
\mathtt{Volume\ of\ cube\ =\ a^{3}}\\\ \\ \mathtt{Volume\ of\ cube\ =\ 20^{3} =\ 8000\ cm^{3}}
So, 8000 cu. cm is the volume of single cube.
Since three cubes are joined next to each other, multiply the above volume by three to get the volume of complete figure.
\mathtt{Volume\ of\ figure\ =\ 8000\times 3=24000\ cm^{3}}
Hence, 24000 cu. cm is the required solution.
Question 04
John bought a cubical water tank of 2 meter side measurement for his house. He decided to cover the side walls of the tank with square tiles of length 25 cm. If the cost of one tile is 3$, find the cost to cover the four walls.
Solution
First calculate the area of four side wall.
Area of one wall = 2 x 2 = 4 sq. meter.
Area of four wall = 4 x 4 = 16 sq. meter.
Hence, the total area to be covered with tiles is 16 sq. meter.
Now find the area of given tile.
Since all the calculation is done in meter, convert the length of tile from cm to meter.
\mathtt{25\ cm\Longrightarrow \ 0.25\ meter}
Area of tile = 0.25 x 0.25 = 0.0625 sq. meter
Now let’s calculate the number of tile used to cover the four side of the wall.
\mathtt{Number\ of\ tile\ =\frac{Total\ area}{Tile\ area}}\\\ \\ \mathtt{Number\ of\ tile\ =\frac{16}{0.0625} =256\ tile}
Hence, total of 256 tiles will be used to cover the four walls.
Now let’s calculate the total cost
Cost of 1 tile = $ 3
Cost of 256 tiles = 256 x 3 = 768$
Hence, the total cost will be 768$
Question 05
If the side of cube is increased by 20%. Find the percentage increase in total surface area of cube.
Solution
Let the side of cube be ” a ” unit.
Surface area of cube = \mathtt{\ 6a^{2}}
Now the side of cube is increased by 20%, new side will be;
\mathtt{\Longrightarrow \ a\ +\ \frac{20}{100} a}\\\ \\ \mathtt{\Longrightarrow \ a+0.2a}\\\ \\ \mathtt{\Longrightarrow \ 1.2\ a}
Hence, new side will be ” 1.2 a ” units.
Surface area of new cube = \mathtt{\ 6( 1.2a)^{2} =8.64\ a^{2}}
Now let’s calculate the percentage increase in surface area.
\mathtt{\%\ increase\ =\frac{8.64a^{2} -6a^{2}}{6a^{2}} \times 100}\\\ \\ \mathtt{\%\ increase\ =\ 44\ \%} \\ \\
So if we increase the side by 20%, the surface area will increase by 44%